Conventional contouring machines which determine the difference in height of points on one surface relative to a second surface typically use an interferometric device. Generally one of the surfaces is being tested against the other reference surface and the height difference is ascertained through determination of the optical path length difference between the surfaces in an interferometer. This determination is made by considering that the intensity at any point is generally represented by the expression:
I.sub.o =.vertline.a.sub.1 .vertline..sup.2 +.vertline.a.sub.2 .vertline..sup.2 +2a.sub.1 a.sub.2 cos (2.pi./.lambda.) .DELTA.(1)
where
(2.pi./.lambda..DELTA. is the phase angle .phi. PA1 .lambda. is the wavelength of the radiation PA1 .DELTA. is the difference in height of the surfaces at corresponding points.
This can be written: EQU I.sub.o =K.sub.o +K.sub.1 cos k.DELTA. (2)
where: EQU K.sub.o =.vertline.a.sub.1 .vertline..sup.2 +.vertline.a.sub.2 .vertline..sup.2 EQU K.sub.1 =2a.sub.1 a.sub.2 EQU k=2.pi./.lambda.
Applying this expression the optical path length may be varied continuously, resulting in the form EQU I.sub.o (t)=K.sub.o +K.sub.1 cos k(.DELTA.+t) (3)
where t is the changing path length. This term may be expanded: EQU I.sub.o (t)=K.sub.o +K.sub.1 cos k.DELTA. cos kt-K.sub.1 sin k.DELTA. sin kt (4)
which is then subject to Fourier analysis. Performance of such analysis requires a very powerful analog computing network or special or general purpose digital computer because of the inherently complex nature of Fourier analysis. In one system over two hundred different intensities I.sub.o are sensed and must be analyzed; and this must be done for each position or spot on the surfaces to be compared. For example, an array of detectors one hundred square requires ten thousand such calculations. Thus an extremely complex task, Fourier analysis, becomes a truly brobdingnagian task used in such applications.
The Fourier analysis produces the first harmonic coefficients: EQU A.sub.1 =K.sub.1 cos k.DELTA. (5) EQU B.sub.1 =K.sub.1 sin k.DELTA. (6)
which are then used to obtain a trigonometric function of the phase angle .phi.=k.DELTA., such as: EQU B.sub.1 /A.sub.1 =-tan k.DELTA. (7)
From this the phase angle is determined and then the difference in height of the surfaces is calculated: EQU .phi.=(2.pi./.lambda.).DELTA. (8) EQU .DELTA.=.lambda..phi./2.pi. (9)
This approach thus requires large, very powerful computing equipment which is expensive and in spite of its size and speed requires much time to complete the computations. These machines must be specially constructed or specially programmed to perform the analysis. The measurement is also time consuming in the case where over two hundred samples of the intensity are made for each position, which takes a minute or more. The extended time required for measurement leads to additional problems: vibrations taking place in the area of the machine interfere with the interferometer operations.